Puzzle for July 21, 2023 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
* AB is a 2-digit number (not A×B).
Scratchpad
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Hint #1
Add B to both sides of eq.2: D - A + B = A - B + C + B which becomes eq.2a) D - A + B = A + C
Hint #2
In eq.3, replace A + C with D - A + B (from eq.2a): E + F - C = D - A + B - D which becomes E + F - C = -A + B Add A and C to both sides of the above equation: E + F - C + A + C = -A + B + A + C which becomes eq.3a) E + F + A = B + C
Hint #3
In eq.1, replace B + C with E + F + A (from eq.3a): E + F + A = A - E + F In the equation above, subtract A and F from each side, and add E to both sides: E + F + A - A - F + E = A - E + F - A - F + E which simplifies to 2×E = 0 which means E = 0
Hint #4
eq.4 may be written as: C = (B + D + E) ÷ 3 Multiply both sides of the above equation by 3: 3 × C = 3 × (B + D + E) ÷ 3 which becomes eq.4a) 3×C = B + D + E
Hint #5
Add A to both sides of eq.2a: D - A + B + A = A + C + A which becomes D + B = 2×A + C which is the same as eq.2b) B + D = 2×A + C
Hint #6
In eq.4a, substitute 2×A + C for B + D (from eq.2b), and 0 for E: 3×C = 2×A + C + 0 which becomes 3×C = 2×A + C Subtract C from each side of the equation above: 3×C - C = 2×A + C - C which makes 2×C = 2×A Divide both sides by 2: 2×C ÷ 2 = 2×A ÷ 2 which makes C = A
Hint #7
Substitute 0 for E, and A for C in eq.3a: 0 + F + A = B + A which becomes F + A = B + A Subtract A from each side of the equation above: F + A - A = B + A - A which makes F = B
Hint #8
Substitute A for C in eq.5: D = A + (A ÷ A) which becomes eq.5a) D = A + 1
Hint #9
eq.6 may be written as: 10×A + B - C = (D × F) - E Substitute A for C, (A + 1) for D (from eq.5a), B for F, and 0 for E in the above equation: 10×A + B - A = ((A + 1) × B) - 0 which becomes eq.6a) 9×A + B = (A + 1) × B
Hint #10
eq.6a may be written as: 9×A + B = (A × B) + (1 × B) which is the same as 9×A + B = (A × B) + B Subtract B from each side of the above equation: 9×A + B - B = (A × B) + B - B which becomes 9×A = A × B Since A ≠ 0 (from eq.5), divide both sides by A: 9×A ÷ A = (A × B) ÷ A which makes 9 = B and also makes 9 = B = F
Hint #11
Substitute 9 for B, and A + 1 for D (from eq.5a), and A for C in eq.2b: 9 + A + 1 = 2×A + A which becomes 10 + A = 3×A Subtract A from both sides of the equation above: 10 + A - A = 3×A - A which makes 10 = 2×A Divide both sides by 2: 10 ÷ 2 = 2×A ÷ 2 which makes 5 = A and also makes 5 = A = C
Solution
Substitute 5 for A in eq.5a: D = 5 + 1 which makes D = 6 and makes ABCDEF = 595609